Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions
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1 Simplifying Rationals 5.0 Topic: Simplifying Rational Expressions Date: Objectives: SWBAT (Simplify Rational Expressions) Main Ideas: Assignment: Rational Expression is an expression that can be written as the quotient of two polynomials ( P ), as long as Q is not 0. Q Activate Finding the domain of a rational expression -If the denominator is 0 of a rational expressions it is said to be undefined..in other words, any value replaced in for the variable that obtains a denominator value of 0 must be excluded from domain h(x) = 2 + 3x 4 2 5x 7 + x 6x + x2 g(x) = 5x p(x) = 9 7x + 5 x 2x 2 + x 3 Simplifying Rational Expressions (Simply put Polynomial Fractions) is looking for Common Factors..Need great FACTORING SKILLS Simplifying Rational Expressions Simplify: 3y(y + 7) (y + 7)(y 2 9) *Just like in fractions, a rational expression is considered undefined when the denominator equals ZERO.so we must exclude all solutions that will created a ZERO in the denominator Original Denominator s Factors (y + 7)(y 2 9) How do we find the values for x that will create a ZERO in the denominator?
2 Simplify and Find the conditions/values that will make the expression undefined. 2x 2 6x 2x x x y y 16 Your Turn x(x + 5) (x + 5)(x 2 16) p 2 + 2p 3 p 2 2p 15 p 2 + 5p + 6 p 2 + 8p + 15 More Examples Simplify: a 4 b 2a 4 2a 3 a 3 b x 4 y 3x 4 3x 3 x 3 y Multiplying and Dividing Find the Product: Find the Quotient: = =
3 Check for Understanding Simplifying Rationals 5.0 8x 21y 3 7y 2 16x 3 3x 15y 5y2 2x 3 3x 3 x 8x 2 5x 5 10mk 2 3c 2 d 5m5 6c 2 d 2 3x 2 y 20ab 6xy 5a 2 b 3 4x 2 5 x3 40 k 3 k k 2 k 2 4k + 3 2d + 6 d 2 + d 2 d + 3 d 2 + 3d + 2 Polynomials x 3 x + 2 x2 + 5x + 6 x 2 9 x 2 + 7x + 10 x 2 + 8x + 15 x 2 + 3x x 2 7x 18 3d + 9 d 2 + 4d + 3 d + 2 d 2 + 5d + 4 x 2 4x x x 2x 2 + 8x + 32
4 Add and Subtract Rational 5.1 Topic: Rational Expressions +and ( ) Date: Objectives: SWBAT (Find LCM of polynomials and ADD and SUBTRACT them) Main Ideas: Assignment: 3b 2 7b + 2 b 2 + 3b 10 2x 2 7x 4 x 2 2x 8 x2 + 7x + 10 x 2 + x 20 Review 3p 2 3p 4p + 4 6p 2 6p p 2 + p Least Common Multiple Finding the LCM: 6, 9, and 12 Find the LCM of 15a 2 bc 3, 16b 5 c 2, and 20a 3 c 6 Your Turn Find the LCM of 6x 2 zy 3, 9x 3 y 2 z 2, and 4x 2 z Find the LCM of 8a 3 bc, 6a 3 b 5 c, and 9a 7 bc 3
5 Polynomials Find the LCM of x 3 x 2 2x and x 2 4x + 4 Find the LCM of x 3 + 2x 2 3x and x 2 + 6x + 9 Find the Sum Adding and Subtracting Polynomials 5a 2 6b a 2 b 2 3x 2 2y xy Check for Understandin g x + 5 3x + 8 2x 4 4x 8 x x x 15 6x 30 x 2 + x x 2 9x x 1 3 x 8 Upper Level
6 Add and Subtract Rational 5.1 Simplifying a Complex Fraction: Method 1 Step 1: Simplify the numerator and the denominator of the complex fraction so that each is a single fraction. Step 2: Perform the indicated division by multiplying the numerator of the complex fraction by the reciprocal of the denominator of the complex fraction. Step 3: Simplify if possible. Simplifying a Complex Fraction: Method 2 Step 1: Multiplying the numerator and denominator of the complex fraction by the LCD of the fractions in both the numerator and the denominator. Step 2: Simplify Complex Fractions 1 a + 1 b 1 b 1 1 a 1 b 1 b + 1 a 3 a 4 b 2 b 1 a
7 x 2 9x 2 4y 2 x 3 2y 3x a 2 a 2 9b 2 a 4 a + 3b 16x 2 25y 2 xy 4 y 5 x 4 5 x + 5 x 5 2 x + 3 x 5
8 Solving Rational Equations 5.2 Topic: Solving Rational Equations Date: Objectives: SWBAT (Solve Rational Equations) Main Ideas: Mixed Review Assignment: Simplify and Find the conditions/values that will make the expression undefined. x 2 49 x State the Domain and Range of 5 x 9 7 Find the Sum. x 1 x + 2 x 2 + 7x + 12 x 2 9 Completely Factor the Polynomial 81x Solve. No short-cuts x = x 3 Back to the Future Now Fraction Bust and Solve. x = x 3
9 Solving Rational Equations Solve. p 2 p 5 p + 1 = p2 7 p 1 + p Check for Understanding Solve. x x 2 = 2 x + 4 2x x 2 + 2x 8
10 Solving Rational Equations 5.2 Solve x = p = 1 p 5 p + 4 p 2 5p More Examples Solve. 5x 20 x 2 9x x 6 = x 4 x 2 9x + 18
11 Solve. 5 n 3 + 5n 2 = 4 n n 2 Last One I Promise (Extraneous Solutions)
12 Reciprocal Functions 5.3 Topic: Graphing Reciprocal Functions Date: Objectives: SWBAT (Determine Properties of Reciprocal Functions and Graph them) Main Ideas: Assignment: 1 x (parent function) Vertex: (0, 0) Parent Graph Shape: Hyperbola Domain: x 0 Range: f(x) 0 Asymptotes Vertical: x = 0 Horizontal: 0 Intercepts: None Vertex Form: a f(b(x c)) + d Limitations on Domain Determine the values of x for which each function is undefined x + 5 3x 2 2 x 2 + 5x 24 6 x 2 3x 28
13 Identify the asymptotes, domain, and range of the function. Check for Understanding Reciprocal Properties Identify the asymptotes, domain, and range of the function. Graph the function 1 x Graph the function 4 x 2 1. Graphing Reciprocal Vertex: Shifts: D: R: Asymptotes: Vertex: Shifts: D: R: Asymptotes: D&R State the Domain and Range of 2 x State the Domain and Range of 4 x + 1 2
14 Can t Touch This! 5.4 Topic: Graphing Rational Functions Date: Objectives: SWBAT (Graph Rational Functions with vertical/horizontal/oblique asymptotes) Main Ideas: Rational Function Asymptotes Assignment: If a(x), a(x) and b(x) are polynomial functions with no common factors b(x) other than 1, and b(x) 0, then: f(x) has vertical asymptotes whenever b(x) = 0 f(x) has at most one horizontal asymptote If the degree of a(x) is greater than the degree of b(x), there is no horizontal asymptote If the degree of a(x) is less than the degree of b(x), then the horizontal asymptote is the line y = 0 (or x axis) If the degree of a(x) is equal to the degree of b(x), then the Examples: horizontal asymptote is the line y = LC of a(x) LC of b(x) A zero of a rational function a(x) occurs at every value of x for b(x) which a(x) = 0 x2 x + 1 One vertical at x = 1 No Horizontal g(x) = 3 x 2 1 Two vertical at x = ±1 Horizontal at y = 0 h(x) = 2x + 1 x 3 One vertical at x = 3 Horizontal at y = 2 Zero at x = 0 No Zeros {3 0} Zeros at x = 1 2 x3 Graph x + 1 x f(x) Graph
15 More Graphing x3 Graph 2x 1 x f(x) A boat traveled upstream at r 1 miles per hour. During the return trip to its original starting point, the boat traveled at r 2 miles per hour. The average speed for the entire trip R is given by the formula R = 2r 1r 2 r 1 +r 2 Draw the graph if r 2 = 15 miles per hour. Application? s a) Graph the function b) What is the R-intercept of the graph? c) What domain and range values are meaningful in the context of the problem? Oblique Asymptote If a(x), a(x) and b(x) are polynomial functions with no common factors b(x) other than 1, and b(x) 0, then f(x) has an oblique (or slant) asymptote if the degree of a(x) minus the degree of b(x) equals 1. The equation of the oblique asymptote is the quotient of a(x) with no remainder. b(x) Example: Vertical Asymptote: x = 1 Oblique Asymptote: x + 3 x4 + 3x 3 x 3 1
16 Can t Touch This! 5.4 x2 Graph x + 1 x f(x) Your Turn Graph x2 3x 10 x 4 x f(x) Point Discontinuation If a(x) b(x), b(x) 0, and x c is a factor of both a(x) and b(x), then there is a point of discontinuity at x = c. Example: Graph x2 4 x 2
17 Name: Rational s Intercepts 5.5 Class: Topic: Intercepts Date: Main Ideas: Review No Calculators Assignment: Which is not a ZERO of the function x 3 3x 2 10x Prove it algebraically a. -3 b. -2 c. 2 d. 4 These are EASY! y-intercept: Where does the graph cross the y-axis? Think about it..when you are on the y-axis, what is the value of x? Y-Intercept It is the same thing as finding f(0): x2 x 6 x 2 1 But always be aware of the RESTRICTIONS ON DOMAIN: x 3 x 2 3x 10 x + 7 x 2 81 Your Turn
18 x-intercepts: Where does the graph cross the x-axis? HEY, this is the same as finding the ZEROS! (You got it DUDE!) Set 0 and solve But, here s the cool thing: For those rational guys, you just need to set the numerator = 0 and solve! WHY? And HOW? Those are some good questions and I am glad you asked X-Intercepts Well, when a fraction is 0, the only way that can happen is if the numerator is 0; So, set the numerator = 0 and solve 0 7 = 0 x2 x 6 x 2 1 But always be aware of the RESTRICTIONS ON DOMAIN: x 3 x 2 3x 10 x + 7 x 2 81 Your Turn
19 Solving Rational Inequalities 5.6 Topic: Solving Rational Inequalities Date: Objectives: SWBAT (Solve Rational Inequalities) Main Ideas: Assignment: Solve. 2x x + 5 x2 x 10 x 2 + 8x + 15 = 3 x + 3 Solving Rational Inequalities Review Steps: Step #1) State the excluded value(s). These are the values that make any of the denominators ZERO. 1 3k + 2 9k < 2 3 Step #2) Solve the related equations. 1 3k + 2 9k = 2 3 Step #3) Use your solution(s) and excluded value(s) to divide a number line into intervals (mainly 3). Step #4) Test values in each interval to determine which intervals contain the values that satisfy the inequality. (Reminder: Don t forget excluded value(s))
20 Solve. 1 x + 3 5x < 2 5 Solve. 3 4 x > 5 4x Upper Level Examples Solve. x 3 1 x 2 < x + 1 4
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